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\smallskip

\noindent {\Large\bf Task D/CCFR-NuTeV/MINERvA} \\
\noindent(CCFR/NuTeV = FNAL Tevatron Experiments E744/E770/E815, NUMI-MINERvA = 
near detector neutrino detector at FNAL)\\
    \\
\noindent{\bf A. Current Participants of the Rochester CCFR-NuTeV Group
                    during 2003-2003}\vspace*{2mm}\\
Faculty: Prof. Arie Bodek, Prof. Kevin McFarland,\\
(Prof. Steve Manly, 
Collaborating Faculty on MINERvA) \\
Research Faculty: Dr. Pawel de Barbaro (CCFR/NuTeV), Dr. Howard Budd 
(CCFR/NuTeV/MINERvA), and\\
                     \hspace*{34mm}Dr. Willis K. Sakumoto 
                     (CCFR/NuTeV/MINERvA)\\ 
		    Graduate Students:  Jesse  Chovjka ((NUMI-MINERvA)
		    REU summer  Undergraduates:  Frank Tompkins, Amee Slaughter 
		    (MINERvA)
		    RET summer High School Teacher: Paul Conrow (MINERvA)
		    Previous Participants: U. K. Yang, S. Avvakumov (PhD Students, CCFR/NuTeV)\\

\noindent{\bf Introduction}\\
\hspace*{0.196in}

\begin{figure}[ht]
\begin{center}
\hspace{.75in}
\epsfxsize=5.0in
\epsfbox{labe.eps}
\end{center}
\caption{ The CCFR/NuTeV Detector }
\label{fig:detector}
\end{figure}

The NuTeV experiment combined an upgraded CCFR
detector (shown in Figure~\ref{fig:detector}), a new continuous
hadron calibration beam, and a new  SSQT neutrino beam. The new SSQT
neutrino beam was sign-selected, thus allowing neutral
current data to be taken separately with neutrinos and
antineutrinos.  The upgraded detector had new liquid
scintillator and new phototubes. Upstream of the detector is a large
volume filled with helium bags. It is a decay channel used in
a search for neutral heavy leptons. The main physics goal of
NuTeV is an improved measurement of the electroweak mixing
angle by using separate neutrino and
antineutrino beams.

The Rochester effort on CCFR/NuTeV in 2002/2003 have focused
on the electroweak mixing angle (led by Prof. McFarland), on
structure functions (led by Bodek, and PhD. Thesis of Yang) and on
limits on oscillatins (led by Bodek, and PhD thesis of Avvakumov).\vspace*{5mm}

MINERvA (McFarland and Morfin co-sppokespersons), is a proposed experiment at 
the near detector hall at NUMI 
to investigate neutrino interactions at low enegies. It can run both
on-axis and off-axis.  The understading and modeling of neutrino
interactions at the low energy range is of fundamental importance to 
both current neutrino oscillations experiments such as K2K and MINOS, 
and essential for future neutrino oscillations experiments such as
JHF to SuperK and NUMI off-axis. The Rochester group is involved
in both of these working groups, and MINERvA is expected to provide
data which is useful to both efforts.

IN 2002-2003 the Rochester effort related to MINERvA has focused on several areas.
(1) McFarland is the co-spokesperson of the MINERvA collaboration.
(2) Bodek and Yang phenomenological work on modeling inelastic electron
and neutrino interactions at low energies. They have extracted 
effective PDFs which work at all Q2, through analysis of previous
electron scattering data. Their method is
is now used in the default Monte Carlo used by K2K, and will provide 
a framework for the analysis of the data in MINERvA.
(23 Bodek and Budd (in collaboration with John Arrington from 
Argonne) have worked on improved modeling of quasielastic electron and
neutrino reactions by doing a re-analysis of previous electron and
neutrino data.  Their work has improved on our knowledge of vector
and axial vector form factors.
(4) Bodek (with the help of Yang, Budd, McFarland and)Manly is now leading an
a collaborative effort between the High Energy and Medium Energy
communities focused on modeling the resonance region in both electron 
and neutrino reactions.
(5) McFarland, Manly and Bodek, with the help of Professor Frank 
Wolfs, two REU undergraduate summer students
and one RET High School Teacher have begun to do research and development
on water based scintillators. Although the primary target at MINERvA is
soid scintillator, the development active water neutrino target is of great interest to
K2K and the JHF to SuperK program.

\noindent{\bf CCFR/NuTeV/MINERvA progress in 2002-2003 and plans for 
future}

\begin{enumerate}
%\hspace*{0.196in} 

\item  Structure Function Analysis \\
\hspace*{0.196in}
The University of Rochester 
is working with the University of Pittsburgh
on the NuTeV structure function analysis. This is a continuation
of the work by Rochester Ph.D. student
Un-ki Yang and Professor Arie Bodek. The Rochester group used CCFR data
to perform the following structure function measurements:
differential cross sections vs $x$ $y$ and $E_{\nu}$, 
$F_2$, $\Delta xF_3 = xF_3^{\nu}$-$xF_3^{\nub}$,
and $R^\nu = {\sigma_L}/{\sigma_T}$.  These were significant
improvements to our knowledge of structure functions and parton distributions.
(Note, additional work on structure functions 
by Yang and Bodek included analysis of electron, 
muon and neutrino scattering,  and CDF W asymmetry data). 

\hspace*{0.196in}
 Dr. Un-Ki Yang
received the Universities Research Association (URA) award 
for the best Fermilab thesis of 2001-2002. As an aside we
tote that
this is the second year in a row that a Rochester PhD student received
this URA Award (a year earlier, the URA Prize was given to
Rochester student Michael Fitch), and the first URA Tollestrup
prize for best postdoctoral work at Fermilab was awarded to
Rochester PhD. Juan Estrada).

Yang's work on CCFR/NuTEV was also recognized by
the Association of Korean Physicists in America, which
presented a citation of Honorable Mention for the 2001
Outstanding Young Researcher Award to Dr. Un-Ki Yang
for his truly outstanding scholarly and pioneering research
in high energy physics. Yang was also awarded the  Lobkowicz prize
for best Ph.D. thesis in high energy physics in the department of physics and
astronomy at the University of Rochester in 2001.

\hspace*{0.196in} The sum of $\nu_\mu$ and $\nub_\mu$
   differential cross sections
for charged current interactions on isoscalar target is related to the
structure functions as follows:
$$
\left[\frac{d^2\sigma^{\nu }}{dxdy}+
\frac{d^2\sigma^{\overline \nu}}{dxdy} \right]
   \frac {(1-\epsilon)\pi}{y^2G_F^2ME_\nu}
     = 2xF_1 [ 1+\epsilon R ] - \frac {y(1-y/2)}{1+(1-y)^2} \Delta xF_3 .
$$
Here $G_{F}$ is the weak Fermi coupling constant, $M$ is the nucleon
mass, $E_{\nu}$ is the incident energy, the scaling
variable $y=E_h/E_\nu$ is the fractional energy transferred to
the hadronic vertex, $E_h$ is the final state hadronic
energy, and $\epsilon\simeq2(1-y)/(1+(1-y)^2)$ is the polarization of virtual
$W$ boson. The structure functions $2xF_1$, $R$, and $\Delta xF_3$
can be extracted from the measurements of $\frac{d^2\sigma^{\nu }}{dxdy}$
and $\frac{d^2\sigma^{\overline \nu}}{dxdy}$
at various values of $\epsilon$.  The structure
function $2xF_1$ is expressed in terms of $F_2$
by $2xF_1(x,Q^2)=F_2(x,Q^2)\times
\frac{1+4M^2x^2/Q^2}{1+R(x,Q^2)}$, where $Q^2$ is the
square of the four-momentum transfer to the nucleon,
    $x=Q^2/2ME_h$ (the Bjorken scaling variable) is
the fractional momentum carried by the struck quark,
and $R={\sigma_L}/{\sigma_T} $
is the ratio of the cross-section of
longitudinally to transversely-polarized $W$-bosons.
The term $\Delta xF_3 = xF_3^{\nu}$-$xF_3^{\nub}$ in leading order
is  $\simeq 4x(s-c)$.

\hspace*{0.196in}
Structure function measurements from NuTeV have several advantages 
over CCFR measurements. Three of these advantages are stated below:
\begin{itemize}
   \item   During the planning stage of NuTeV, Prof Arie Bodek
      conceived of the continuous hadron and muon calibration beam. In CCFR, the 
      calibration beam entered the side
        of the building, and we had to 
      move the detector and disrupt
      data taking to do calibrations. 
       The hadron beam line was modified 
       and redirected at the cener of the building.
        NuTeV ran the calibration beam
      concurent with neutrino data taking. 
      The Rochester group analyzed the hadron energy calibration 
      data which resulted in
       a smaller error of 0.43\% (versus 1.0\% in CCFR).  Note that
        Rochester was also responsible for the hadron energy 
        calibration for CCFR.
   \item  Since the NuTeV beam
      is sign selected, even events for 
      which the final state muon is not analyzed by the toroid 
      spectrometer can be used in structure
      function analysis (because of the separate running in 
      $\nu$ and $\overline{\nu}$ modes). 
      These "range outs" extend the range to high $y$ and low $\epsilon$.
   \item   Since the  sign selected beam creates either $\nu$ or
      $\overline{\nu}$, the final
      state muons are always focused 
      toward the center of the muon spectrometer.
      This results in higher acceptance for  high $y$ events with
      low energy final state muons.
\end{itemize}


\begin{figure}[ht]
\begin{center}
\epsfxsize=4.5in
\epsfbox{f2q2lowx.eps}
\end{center}
\caption{ NuTeV preliminary $F_2(x,Q^2)$. NLO curve is 
TR-VFS with MRTS99 (Thorne \& Roberts Phys.Lett B421,303(1998)) }
\label{f2q2lowx}
\end{figure}

\hspace*{0.196in}
the CCFR structure function results
were published. Preliminary results of the NuTeV cross sections 
and structure functions, $F_2$, $xF_3$, and  $R$
were presented in 2003. Figure~\ref{f2q2lowx} shows the 
NuTeV preliminary $F_2(x,Q^2)$. 
We see agreement with the previous CCFR $\nu$-Fe measurements. In 
addition, we see good agreement with the NLO QCD predictions. 
The preliminary NuTeV measurement of R and $xF_3$ is also in good agreement
with the previous CCFR results. 

\hspace*{0.196in}
The plan is to finalize the NuTeV cross section 
and structure functions in the latter part
of 2003. The NuTeV "range-out" data, which
measure the differential cross sections at high $y$, will be added 
to the sample. This will give a better $2xF_1$,  $R$ fit.
The same type of QCD analysis which CCFR performed will be done with
the NuTeV structure functions. This includes the QCD fits to PDFs
and extraction of $\alpha_s$. 

     \item  Precision Determination of  $\sin^2 \theta_W$ \\
\hspace*{0.196in} Professor Kevin McFarland has been leading the CCFR
and NuTeV analysis of $\sin^2 \theta_W$.

\hspace*{0.196in} Neutrino-nucleon scattering is one of
the most precise probes of the weak
neutral current. The ratio of neutral current (NC) to charged current (CC)
cross-sections
for either $\nu$ or $\nub$ scattering
from isoscalar targets of $u$ and $d$ quarks can be written as:
%\cite{llewellyn}
$$
R^{\nu(\nub)} \equiv \frac{\sigma(\nunub N\rightarrow\nunub X)}
                   {\sigma(\nunub N\rightarrow\ell^{-(+)}X)}
= (g_L^2+r^{(-1)}g_R^2), \mbox{ where }
r \equiv \frac{\sigma({\overline \nu}N\rightarrow\ell^+X)}
                  {\sigma(\nu N\rightarrow\ell^-X)} \sim \frac{1}{2},
$$
where  $g_{L,R}^2$ are the average effective left and right-handed 
$\nu$-quark coupling.
Charm production, which affects CC cross-sections,
is a major theoretical uncertainty. Hence,
NuTeV measures the Paschos-Wolfenstein variable
$$
R^{-} \equiv \frac{\sigma(\nu_{\mu}N\rightarrow\nu_{\mu}X)-
                     \sigma(\nub_{\mu}N\rightarrow\nub_{\mu}X)}
                    {\sigma(\nu_{\mu}N\rightarrow\mu^-X)-
                     \sigma(\nub_{\mu}N\rightarrow\mu^+X)}
= \frac{\Rnu-r\Rnub}{1-r}=(g_L^2-g_R^2)
= \frac{1}{2}-\sin^2 \theta_W.
$$
Measuring R$^-$ required a new beam line to create separate $\nu$ and
$\nub$ beams.
This new NuTeV SSQT
beam also reduces the uncertainty in the other major
systematic error, including the $\nue$ contamination in the beam.
In addition, the continuous calibration of
the detector reduces the detector related systematic errors.

\hspace*{0.196in} NuTeV measures the ratio of short to long events in
the $\nu$ and $\nub$ beams to be:
$$\Rmeasnu= 0.3916 \pm 0.0007~{\rm and}~\Rmeasnub= 0.4050 \pm 0.0016.$$
A detailed leading order (LO) Monte Carlo program converts \Rmeasnu
and \Rmeasnub to $ \sin^2\theta_W$.
This yields:
      $\sin^2\theta_W^{({\rms on-shell)}}=0.2277\pm0.0016$,
assuming $M_{top}$=175 GeV and $M_{Higgs}$=150 GeV.
This is in very good agreement with the world average of all
previous neutrino experiments of
$\sin^2 \theta_W=0.2277 \pm 0.0036$ (but the NuTeV errors are smaller).
The leading terms in the one-loop electroweak radiative corrections
%\cite{bardin}
produce the small residual dependence of our result on M$_{top}$ and
M$_{Higgs}$.  The Standard Model fit to  all
other electroweak measurements  excluding neutrino experiments gives
$\sin^2 \theta_W=0.2227\pm0.00037$, approximately $3\sigma$ from the NuTeV result.

\hspace*{0.196in} The NuTeV result has generated a great deal 
of interest. If new physics is the explanation, 
then the NuTeV result requires new physics at the tree 
level. This is difficult to accomplish within
the various existing "natural" models for Physics beyond the 
Standard Model.  Hence, much of the discussion on the NuTeV 
anomaly has been on Standard Model explanations.
Our recent efforts have been devoted to the investigation of
possible conventional explanations for the 3 $\sigma$ deviation from the 
Standard Model. These include the NLO and NNLO QCD corrections,
PDF uncertainties, charm uncertainties, isospin breaking effects, and nuclear 
effects. Many of these effects are very small for $R^-$.
However, NuTeV does not measure $R^-$ over the entire kinematic range. Note
that $R^-$ is the ratio of cross sections,
while NuTeV measures the ratio of events with exprimental cuts. 
The difference between $R^-$ and NuTeV's measurement includes the 
experimental cuts on ($E_{had}$), backgrounds, cross talk between 
NC and CC, differences between the NC and CC acceptance, charm production, etc. 
 
\hspace*{0.196in} The QCD corrections to DIS neutrino scattering for NLO and 
NNLO are known. The Paschos-Wolfenstein ratio can be written 
to include these higher order terms. This results in 
a NLO QCD correction to $R_{model}^-$=-0.00033
(which brings the value
about 1/4$\sigma$ closer to the SM prediction). This
correction includes the low y-cut from the 20 
GeV $E_had$ cut, and  an effective high y-cut from 
being unable to see very low energy muons in CC events.
  
\hspace*{0.196in} It has been suggested that the NuTeV
results might be explained  by an asymmetric strange
sea, i.e. if $\sav\neq\savbar$.  However, this option has
been ruled out in our recent publication, PRD 65:111103,2002 
(hep-ex/0203004). Here we use the NuTeV opposite
sign dimuon measurement to extract the quantity
$\Sav-\Savbar = -0.0027 \pm 0.0013$.
Instead of explaining the NuTeV results, our measurement
of the strange sea asymmetry results in an
increase in the NuTeV value of $\stw$,
   $\Delta\stw = +0.0020 \pm 0.0009$, which increases the discrepancy
   with respect to
Standard Model expectation to $3.7\sigma$.

The possibility of an asymmetric strange sea comes from 
fact that the CDHSW structure functions for neutrino and
antineutrinos are inconsistent with each other at large x.
The CDHSW structure functions deviate from the QCD prediction at 
high x, and an asymmetric strange sea is given as a
possible explanation.  However, we note that
the CCFR structure functions do not show this effect and
are consistent with QCD. Therefore there is no 
evidence for an  asymmetric strange sea from the CCFR structure 
function data, and a limit has been placed on this asymmetry
using the dimuon data.
A preliminary analysis of the CCFR
data for the $s$ and $\overline{s}$ asymmetry
within an NLO cross-section model indicates that
the momenta carried by the strange and antistrange seas 
are consistent with each other within 
experimental uncertainties. 

To best answer these and other questions, we are building a full 
NLO $\nu$ DIS event generator. However,  ee do not believe this will 
be a big effect.  This is a calculation to order
$\alpha_s$ based on a 1978 paper of Altarelli, Ellis, Martinelli.
As $R_L$ is included in our Monte Carlo program, we only need to include
corrections to $xF_3$. The plan will be to do light quarks 
first and then put in NLO charm production. 
A NuTeV thesis topic (D. Mason, University of Oregon) 
is a NLO analysis of charm production,
which includes an NLO study of the strange sea. 
This code will included in our Monte Carlo program for 
$\stw$.

\begin{figure}[htp]
\centerline{
\psfig{figure=osclimit_combo_color.eps,width=3. in,height=6.0in}}
\caption{(a) Excluded region of $\sin^2 2\alpha$ and $\delta m^2$ for
$\nu_\mu \rightarrow \nu_e$ oscillations from the NuTeV analysis at 90\%
confidence is the area to the right of the dark, solid curve. (b) NuTeV
limits for $\overline\nu_\mu \rightarrow  \overline\nu_e$. (c)
Combined NuTeV limits for
$\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$,
assuming oscillation parameters for $\nu$ and
     $\nub$ are the same.  }
\label{fig:osc}
\end{figure}

\item  Search for $\nu_\mu(\overline\nu_\mu)$ Oscillations \\
\hspace*{0.196in} S. Avvakumov, a Rochester PhD student
on NuTeV (PhD 2002), extracted
limits on $\nu_\mu \to \nu_e $ and
$\overline{\nu}_\mu \to \overline{\nu}_e$
oscillations for his Ph.D. thesis.
His thesis, done under the supervision of
Professor Arie Bodek, was submitted in Jan 2002 . The results
were published in PRL 89 011804,2002.
(S.~Avvakumov {\it et al.},
A search for $\nu_\mu\to\nu_e$ and $\nubar_\mu\to\nubar_e$
oscillations  at NuTeV (hep-ex/0203018)).

\hspace*{0.196in} Since NuTeV had separate
$\nu_\mu$ and $\overline{\nu}_\mu$ beams,
we  search for oscillations in both running modes, without the need
to assume that the oscillations parameters for neutrinos and
antineutrinos are the same.
The oscillations are searched for using a statistical
separation of $\nu_e N$ charged
current interactions in the  NuTeV detector at Fermilab.
The $\nu_e$ interactions  are identified by the difference in the
longitudinal shower energy deposition pattern of $\nu_e N \rightarrow
eX$ versus $\nu_\mu N \rightarrow \nu_\mu X$ interactions.
Using this technique,  the absolute flux of $\nu_e$'s
at the detector is measured and is compared to the flux predicted by a detailed
beam line simulation. Any excess could be interpreted as a
signal of $\nu_\mu \rightarrow \nu_e$ oscillations.



\hspace*{0.196in}
At all $\Delta m^2$, the data are consistent with no
observed $\nu_\mu \rightarrow \nu_e$ oscillations.
The frequentist approach
is used to set a 90\% confidence
upper limit for each $\Delta m^2$. The limit in $\sin^2 2\alpha$
corresponds to a shift of 1.64 units in $\chi^2$ from the minimum.
The 90\% confidence upper limit is shown in Fig. \ref{fig:osc}(a) for
$\nu_\mu \rightarrow \nu_e$. Also shown are limits from
BNL-E734 and BNL-E776.
For $\sin^2 2\alpha = 1$, $\Delta m^2 > 2.4 $~${\rm eV^2}$ is excluded, and for
$\Delta m^2 \gg 1000$~${\rm eV^2}$, $\sin^2 2\alpha > 1.6 \times
10^{-3}$. In the large $\Delta m^2$
%and small $\sin^2 2\alpha$
region, NuTeV provides improved limits for  $\nu_\mu
\rightarrow \nu_e$ oscillations.

\hspace*{0.196in}
Similarly, the limit for  $\overline\nu_\mu \rightarrow  \overline\nu_e$
is shown Fig. \ref{fig:osc}(b). Also
shown are the LSND results and preliminary results from KARMEN.
For the case of $\sin^2 2\alpha = 1$,
$\Delta m^2 > 2.6 $~${\rm eV^2}$ is excluded, and for
$\Delta m^2 \gg 1000$~${\rm eV^2}$, $\sin^2 2\alpha > 1.1 \times
10^{-3}$.  In  the $\nub_\mu$ mode,
our results
exclude  the high $\Delta m^2$
end of $\overline\nu_\mu \rightarrow  \overline\nu_e$
oscillations parameters favored by the LSND experiment,
without the need
to assume that the oscillation parameters for $\nu$ and
   $\nub$ are the same.
    These
are the most stringent experimental limits %~\cite{karm}
for $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$
   oscillations in the large $\Delta m^2$ region.

\hspace*{0.196in}
If we assume that the oscillation parameters for $\nu$ and
   $\nub$ are the same, we can
     combine our $\nu$ and $\nub$ results and compare to the CCFR results
     with a mixed beam.
     The combined NuTeV results exclude
      $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$
   oscillations with $\sin^2 2\alpha > 0.9 \times 10^{-3}$ for
   large  $\Delta m^2 \gg 1000$~${\rm eV^2}$. For $\sin^2 2\alpha =
   1$, $\Delta m^2 > 2.2 $~${\rm eV^2}$ is excluded.  These
   are the most stringent experimental limits for
   $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$
   oscillations in the large $\Delta m^2$ region.

\end{enumerate}

%------

\noindent{\bf Neutrino Oscillations and Electron and Neutrino Scattering 
at Low Energies}

Arie Bodek, Howard Budd, Kevin McFarland (in collaboration with
Prof. Steve Manly)\\


The recent discoveries of neutrino oscillations in atmospheric
neutrinos~\cite{SK} and in neutrinos from the sun~\cite{SNO,KAMLAND}
motivate the detailed studies of neutrino oscillations at future high
intensity neutrino beams from accelerators.  The two disparate mass
scales observed in oscillations from these astrophysical sources,
$\dmsq_{\rms atm}\approx \evsq{2\times10^{-3}}$ and $\dmsq_{\rms solar}\sim
\evsq{10^{-4}}$, along with the stringent limits on $\nuebar$
disappearance at the atmospheric $L/E$ in the CHOOZ and Palo Verde
reactor experiments experiments\cite{CHOOZ,PaloVerde}, have raised the
possibility that there may be an observable CP-asymmetry in
$\numu\to\nue$ transitions.  This rare, sub-leading transition in the
neutrino flavor sector is analogous to to searching for first and
third generation mixing in the quark sector, which has led to a rich
phenomenology of CP-violation, meson mixing and rare decays in the
quark sector.

These neutrino oscillation experiments are very challenging, because
of the required $L/E$ of $400$~km/GeV. The experiments
 require megawatt proton
sources, $1-3$~GeV neutrino beams and multi-kiloton detectors to make
the observations.  The measurements are further complicated by the low
transition probability of $\numu\to\nue$ and the need to compare to
$\numubar\to\nuebar$ at high precision.  This requires a detailed
knowledge of the neutrino interaction cross-sections both for the
dominant signal processes and for background processes, such as
$\nu N\to\nu N \pi^0$ where the $\pi^0$ is misidentified as an
electron in a many kiloton sampling detector.  

\begin{figure}
    %[t]
    \begin{center}
\epsfig{figure=Fig1Mex.eps,width=6.0in,height=6.0in}
\caption{Electron and muon $F_2$ data (SLAC, BCDMS, NMC, H1 94)
used to obtain the parameters of the Bodek-Yang
modified GRV98 $\xi_w$ fit compared to the predictions of the unmodified GRV98
PDFs (LO, dashed line) and the modified GRV98 PDFs 
fits (LO+HT, solid line); [a] for $F_2$ proton, [b] for $F_2$ deuteron,
and [c] for the H1 and NMC proton data at low $x$.}
\label{fig:f1fit}
\end{center}
\end{figure}

\noindent{\bf Modeling Electron and Neutrino Scattering 
at Low Energies in the Continuum Region}

A. Bodek (in collaboration with U. K. Yang)

The phenomenology of neutrino cross-sections is relatively simple when
$E\nu\ll 1$~GeV or when $E\nu\gg$~few GeV since these regimes are
dominated by (quasi)-elastic and deep inelastic processes,
respectively.  However in the $1$ to few GeV region, there are
contributions to the cross-section from both of these processes as
well as resonance-dominated hadroproduction.  A successful
phenomenological approach to modeling the resonance region in electron
scattering is the use of quark-hadron duality to relate quark-model
cross-sections to the cross-section over the discrete
resonances~\cite{bodek-yang3} as shown in Fig.~\ref{fig:f1fit},
Fig.~\ref{fig:predict} and Fig.~\ref{fig:predictD}.
 Figure~\ref{fig:f1fit} shows a fit by Bodek and Yang to inelastic
 electron and muon scattering data with a modified scaling variable
 and GRV98 PDFs with additional corrections (based on consideration
of the Adler and Gilman sum rules~\cite{Adler}. 
Figure~\ref{fig:predict} and Figure~\ref{fig:predictD} compare
the predictions of the fit to data in the resonance region
(which is not included in the fit, as well as other data
%UPDATE
such a photoproduction and high energy neutrino data).  All 
predictions assume quark-model relations, and an empirical
fit to R ($R_{1998}$).
This approach requires the separation
of the $F_{2}$, which has a simple representation in the quark
model, and $R$ whose description requires a different
prescription.  We plan to build successful models of neutrino
scattering using this same prescription with the addition of a
quark-model representation of the axial vector component of the
cross-section.

Since neutrino data are
measured in nuclear targets, even in 
the quark model case,  the separated vector structure functions 
from electron scattering,  $F_{2p}$, $F_{2n}$, $R_{p}$, $R_{n}$
for bound nucleons are needed in order 
to understand the axial structure function
in neutrino scattering experiments.
%


\begin{figure}
    %[t]
    \begin{center}
\epsfig{figure=Fig2Mex.eps,width=6.0in,height=6.0in}
\caption{ Comparisons to proton and
iron data not included in the Bodek-Yang GRV98 $\xi_w$  fit. 
(a) Comparison of SLAC and JLab 
(electron) $F_{2p}$ data
in the resonance region (or fits to these data)
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) the  Bodek-Yang modifications.
(b)  Comparison of photoproduction
       data on protons  to predictions using
 Bodek-Yang modified GRV98 PDFs. 
(c)  Comparison of representative CCFR $\nu_\mu$ and $\overline\nu_\mu$
charged-current differential cross sections 
on iron at 55 GeV and the predictions of the GRV98 PDFs with (LO+HT,
solid) and without (LO, dashed)
       Bodek-Yang modifications.  }
\label{fig:predict}
\end{center}
\end{figure}

\begin{figure}
    %[t]
    \begin{center}
\epsfig{figure=
Fig3Mex.ps,width=5.5in,height=3.2in}
\caption{ Comparisons to data on deuterium
which were not included in the  Bodek-Yang GRV98 $\xi_w$ fit. 
(a) Comparison of SLAC and JLab 
(electron) $F_{2d}$ data
in the resonance region 
and the predictions of the GRV98 PDFs with (LO+HT, solid)
and without (LO, dashed) our modifications.
(b)  Comparison of photoproduction
       data on deuterium  to predictions using
Bodek-Yang modified GRV98 PDFs (including shadowing corrections). 
(c) The  shadowing corrections that were applied to the PDFs for
predicting the photoproduction cross section on deuterium.}
\label{fig:predictD}
\end{center}
\end{figure}


\noindent{\bf Modeling Electron and Neutrino Scattering 
at Low Energie in the Resonance Region}

Work begun in collaboration with Cynthia Keppel of
Hampton University and Jefferson Laboratory.



In addition to our investigation
of the inelastic continuum we plan to use data taken
by the Rochester group at SLAC~\cite{e140}
new precise data from Jefferson Lab~\cite{ejlab} to do a combined analysis
of electron-nucleon and neutrino nucleon data in the resonance region
within the Feynman quark-oscillator model as done
years ago (with poor precision)  by Rein and Seghal~\cite{rs}.
The results
of the updated Rein-Seghal type of analysis will be compared to an analysis
which is based on duality.

The full program of studies will use precise electron
scattering data, in particular $\sigma_L$ and $\sigma_T$ 
%UPDATE
(or equivalenty $F_{2}$ and $R$)
%
on Hydrogen and Deuterium~\cite{ejlab}. These studies will later be supllemented with
data on nuclear targets
targets (materials suited for future neutrino oscillation detectors --
water \cite{jhf}, hydrocarbons \cite{numi}, liquid argon -- and steel,
where the most precise high energy neutrino cross-sections have been
measured \cite{CCFR}) in the relevant kinematic regime.  Later, as the
new generation of high rate neutrino beams at Fermilab and J-PARC become
available, the approach can be directly validated with
comparisons to data from high rate neutrino cross-section experiments
on the same targets \cite{eois}.





%
\noindent{\bf Modeling Quasi-elastic Form Factors for Electron and Neutrino Scattering}

Arie Bodek, Howard Budd (in Collaboration with John Arrington of 
Argonne National Laboratory). This work is ongoing and has only 
been presented in conference 
proceedings late 2002 and 2003. It is being written up for publication.

Since Quasielastic scattering forms an important component of 
neutrino scattering
at low energies, we have undertaken to investigate 
QE neutrino scattering using the latest information. 

Recent experiments at SLAC and Jefferson Lab (JLAB) have 
given very precise measurements of the vector electro-magnetic
form factors for the proton and the neutron. 
These form factors can be related to the form factors for QE
neutrino scattering by conserved vector current hypothesis, CVC. 
These more recent form factors can be used to give better predictions
of QE scattering.

The hadronic current for QE neutrino scattering is given by
$$ <p(p_2)|J_{\lambda}^+|n(p_1)> = 
 \overline{u}(p_2)\left[
  \gamma_{\lambda}F_V^1(q^2)
  +\frac{i\sigma_{\lambda\nu}q^{\nu}{\xi}F_V^2(q^2)}{2M} 
  +\gamma_{\lambda}\gamma_5F_A(q^2)
  +\frac{q_{\lambda}\gamma_5F_P(q^2)}{M}
  \right]u(p_1) 
$$
We do not include second class currents, so the scaler
form factor  $F_V^3$ and the tensor form factor $F_A^3$
are not included. 
Using the above current, the cross section is

$$ \frac{d\sigma^{\nu,~\overline{\nu}}}{dq^2} = 
  \frac{M^2G_F^2cos^2\theta_c}{8{\pi}E^2_{\nu}}
  [A(q^2) \mp \frac{(s-u)B(q^2)}{M^2} + \frac{C(q^2)(s-u)^2}{M^4}],
$$
where
%$$ s-u = 4ME_{\nu} + q^2 - m_l^2$$
\begin{eqnarray*}
%A(q^2) & = & 
A(q^2)= 
 \frac{m^2-q^2}{4M^2}\left[
    \left(4-\frac{q^2}{M^2}\right)|F_A|^2
  -\left(4+\frac{q^2}{M^2}\right)|F_V^1|^2
  -\frac{q^2}{M^2}|{\xi}F_V^2|^2\left(1+\frac{q^2}{4M^2}\right) 
  -\frac{4q^2ReF_V^{1*}{\xi}F_V^2}{M^2} 
    \right],
%\right.  \\    &   & \left.
%  -\frac{m_l^2}{M^2}\left(|F_V^1+{\xi}F_V^2|^2
%    \right)
\end{eqnarray*}
$$B(q^2) = -\frac{q^2}{M^2}ReF_A^*(F_V^1+{\xi}F_V^2), $$
$$
C = \frac{1}{4}\left(|F_A|^2 + |F_V^1|^2 -
 \frac{q^2}{M^2}\left|\frac{{\xi}F_V^2}{2}\right|^2\right).
$$
We have not shown terms in $(m_l/M)^2$, and  $F_P(q^2)$
is multiplied by  $(m_l/M)^2$. (Note, $F_P(q^2)$ is included
in the calculations.)
The formulas for $ F^1_V(q^2)$ and  $cF^2_V(q^2)$
are
$$ F^1_V(q^2)=
\frac{G_E^V(q^2)-\frac{q^2}{4M^2}G_M^V(q^2)}{1-\frac{q^2}{4M^2}},~~~
{\xi}F^2_V(q^2) =\frac{G_M^V(q^2)-G_E^V(q^2)}{1-q^2/4M^2} $$

We use the CVC to determine $ G_E^V(q^2)$ and $ G_M^V(q^2)$ 
from  the electron scattering form factors
$G_E^p(q^2)$, $G_E^n(q^2)$, $G_M^p(q^2)$, and $G_M^n(q^2)$.

$$ G_E^V(q^2)=G_E^p(q^2)-G_E^n(q^2),~~~G_M^V(q^2)=G_M^p(q^2)-G_M^n(q^2) $$

Many of the neutrino experiment have assumed the 
form factors are the dipole approximation.
$$ G_D(q^2)=\frac{1}{(1-q^2/M_V^2)^2},~~~M_V^2=0.71~GeV^2$$
$$ G_E^p=G_D(q^2),~~~G_E^n=0,~~~G_M^p={\mu_p}G_D(q^2),~~~ 
G_M^n={\mu_n}G_D(q^2)
$$

The axial form factor is given by 
$$ F_A(q^2)=\frac{g_A}{(1-\frac{q^2}{M_A^2})^2 } $$
This form factor needs to be determined from 
QE neutrino scattering. 
Older experiments used $g_A=-1.23$, however 
the current value is -1.267.
The world average from neutrino experiments  is
$M_A$ = 1.026 $\pm$ 0.02 GeV. The value of $M_A$ depends 
on the electro-magnetic form factors. 
Since we are updating these form factors, we need to 
determine a new value of $M_A$ using these latest
form factors and $g_A$. $M_A$ can also 
be determined from pion electro-production,
which gets 1.069 $\pm$ 0.016 GeV (it is expected
that this determination is not as reliable as
that from neutrino data because of theoretical
corrections. These corrections bring the value
into closer agreement with the value as measured
in neutrino reactions).

From PCAC, the pseudoscaler form factor $F_P$ is
$$ F_P(q^2)=\frac{2M^2F_A(q^2)}{M_{\pi}^2-q^2}. $$
$F_P(q^2)$ is multiplied by  $(m_l/M)^2$ so its effect 
is very small except at very low energy, \(< .2 GeV\).

\begin{figure}
\begin{center}
\epsfxsize=3.01in
\mbox{{\epsffile{show_gep_ratio.eps}}\epsfxsize=3.01in{\epsffile{show_gmp_ratio.eps}}}\\
\end{center}
\caption{Our fits for $G_E^p/G_D$ and  $G_M^p/{\mu}_{p}G_D$.
The fits with and without the polarization measurements are shown.
Polarization data is shown in cyan.}
\label{show_gep}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=4.01in
\mbox{\epsffile[79  408  549  720]{show_gepgmp_ratio.eps}}
\end{center}
\caption{ Ratio of $G_E^p$ to $G_M^p$ as extracted by Rosenbluth
measurements and from polarization measurements.  }
\label{gegm_proj}
\end{figure}

We have used almost all data from 
SLAC and JLAB to determine the form factors.
Form factors can be determined by using 
older technique of Rosenbluth separation
(cross section) or the newer technique of polarization transfer 
from JLAB. Figure~\ref{show_gep} shows the ratio of 
our fits divided by the dipole, $G_D$. The JLAB polarization 
measurement does not directly measure the form factors,
but measures the ratio $G_E^p$/$G_M^p$. As we see from 
figure~\ref{gegm_proj},  $G_E^p$/$G_M^p$
is flat vs $Q^2$ ($Q^2=-q^2$) for the cross section measurement. 
However, $G_E^p$/$G_M^p$ decreases for the polarization 
measurement. Although the polarization measurement is
believed to have smaller systematic error especially 
at high $Q^2$, the origin of this disagreement is not known.
Experiments at JLAB hope to resolve this disagreement. 

We fit  electron scattering data to an inverse polynomial
$$ Poly^{-1}(Q^2)=\frac{1}{1+a_2Q^2+a_4Q^4+a_6Q^6+...}. $$
Table~\ref{JRA_HallA_coef} shows the results of our fit. 
These fits uses both cross section data 
and polarization transfer data from JLAB. In addition,
we have fits which just use the cross section data.

\begin{table} 
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\noalign{\vspace{-8pt}} \hline
     & $a_2$ & $a_4$ & $a_6$ & $a_8$ & $a_{10}$ &  $a_{12}$  \\ \hline
$G_E^p$ & 3.253 & 1.422 & 0.08582 & 0.3318    & -0.09371 & 0.01076 \\
$G_M^p$ & 3.104 & 1.428 & 0.1112  & -0.006981 & 0.0003705 & -0.7063E-05 \\
$G_M^n$ & 3.043 & 0.8548& 0.6806  & -0.1287   & 0.008912  &       \\ \hline
\end{tabular}
\end{center}
\caption{ The coefficients of the inverse polynomial for the 
$G_E^p$, $G_M^p$, and $G_M^n$. This fit uses both cross section data 
and polarization data from electron scattering. }
\label{JRA_HallA_coef}
\end{table}


Previous experiments assumed  $G_E^n(q^2)$ = 0. 
Since the neutron has no charge, $G_E^n(q^2)$ must be
zero at $q^2$ = 0. However, it doesn't have to zero for 
$q^2 \neq$ 0. New JLAB polarization transfer data gives
a precise non-zero value of $G_E^n(q^2)$.
Our analysis uses  $G_E^n(q^2)$ 
from Krutov et. al. (Hep-ph/0202183).
$$G_E^n = -\mu_n\frac{a\tau}{1-b\tau}G_D(q^2),~~~\tau=\frac{Q^2}{4M^2}.$$

\begin{figure}
\begin{center}
\epsfxsize=3.01in
\mbox{{\epsffile[7 5 550 400]{ratio_JKJJ_D0DD.eps}}
\epsfxsize=3.01in{\epsffile[7 5 550 400]{ratio_JhaKJhaJ_D0DD.eps}}}\\
\end{center}
\caption{ Ratio of cross section vs energy using the most updated form
factors vs the dipole approximation. The left plot uses the Rosenbluth 
separation data and the polarization data. The right plot uses
Rosenbluth separation data.  }
\label{ratio_JKJJ_D0DD}
\end{figure}

Figure~\ref{ratio_JKJJ_D0DD} shows the ratio of the QE cross section
using the most updated form factors vs the dipole approximation. 
The most updated form factor are our fits to the form factors for
$G_M^p$, $G_E^p$, and  $G_M^n$ and Krutov $G_E^n$. 
The left plot uses the cross section 
data and the polarization data. The right plot uses
Rosenbluth separation data. The cross section for neutrino QE 
scattering is independent of the polarization data. 
The differences between the polarization 
data and the cross section data are at high $Q^2$, 
while the form factors contribute to the cross section at low $Q^2$.

  There is a big effect in the cross section between using the latest
form factors or not. The difference is 3\% at high energy and can become
as much as 6\% at 1 GeV. Figure~\ref{ratio_JKJJ_D0DD} shows the difference
between $G_E^n$ = Krutov vs $G_E^n$ = 0. We see all of the difference
at high energy and most of the the difference at low energy is due
to $G_E^n$. At low energy, which are the energies for neutrino 
oscillation experiments, the other form factors are important.  

\begin{figure}
\begin{center}
\epsfxsize=3.01in
\mbox{{\epsffile[7 5 550 400]{ratio_DKDD_D0DD.eps}}
\epsfxsize=3.01in{\epsffile[7 5 550 400]{ratio_Jha0JhaJ_D0DD.eps}}}\\
\end{center}
\caption{ Ratio of cross section vs energy using different
sets of form factors vs energy. The left plot looks at 
the difference between using $G_E^n$ = Kurtov vs $G_E^n$ = 0.
The right plots looks at the difference between using 
our fits for $G_E^p$, $G_M^p$, and $G_N^n$ vs the dipole
approximation.
 }
\label{ratio_DKDD_D0DD}
\end{figure}

  A 1\% increase in either $M_A$ or $|g_A|$ increases the
cross section about 1\%. As the old value of $g_A$=-1.23.
and more recent values of -1.267 increases the cross section 
by about 2.5\%. In addition the more recent value of $M_A$
of 1.02 vs the 1.032 causes the cross section to fall by about 1\%.
$M_P$ has almost no effect on the cross section except at very
low $E_{\nu}$.

  Previous neutrino experiment, mostly bubble chambers, extract
$M_A$ using the best known assumptions at the time. Changing
these assumptions changes $M_A$. Hence, we
 use published data  to extract
a corrections to $M_A$ using our form factors.
They give their data in histograms of corrected events.
Their flux is shown in figures, which we 
parameterize using  a spline fit. 
We calculate the $Q^2$ distribution of their data
and fit their data  for $M_A$.
We determine $M_A$ using their assumptions and our assumptions. 
Figure~\ref{Baker_ma107_nor99} shows a histograms of the $Q^2$ distribution
for both Baker et. al. and Kitagaki et. al. Our 
curves agree very well with their curves. In addition our curves 
agree very well with Barish et al, but not quite so well with 
Miller et al. As Miller gives the final result of 
Barish, adding 3 times the data, they should be using the same 
code. Therefore, the discrepency between Miller and Barish
is puzzling.     


\begin{figure}
\begin{center}
\epsfxsize=3.01in
\mbox{{\epsffile[124  208  538  587]{Baker_81_ma107_nor100.eps}}
\epsfxsize=3.01in{\epsffile[124  208  538  587]{Kit_83_105_flux_spline_area.eps}}}\\
\end{center}
\caption{ $Q^2$ distribution from Baker et. al. and Kitagaki et. al. 
The red curve is our calculation using their assumptions. The
blue curve is their calculation taken from their $Q^2$ distribution histogram.}
\label{Baker_ma107_nor99}
\end{figure}

  We fit for $M_A$ using their assumptions and our assumptions, and 
we determine the shift using our assumption. They calculate
$M_A$ using unbinned maximum likelihood, which we can't do since
we do not have the events. We use binned maximum likelihood.
These experiment use a dipole correction from Ollson et. al.
[PRD 17 2938 (1978)]. Table~\ref{MA_values} gives the 
result of the calculation. We also determine $M_A$ using the dipole. 
We agree with the value of $M_A$ of Baker, 
but disagree with the values of Barish, Miller, and Kitagaki.
However, as previously stated our $Q^2$ distributions agree
very well with Baker, Barish, and Kitagaki. Maybe
their unbinned likelihood fit as opposed to our binned
likelihood fit creates the difference. But then why 
did we get Baker's value correct? We do not 
have an explanation for the discrepency in $M_A$.
The table indicates we should shift
the value of $M_A$ determined from  deuterium down by 0.025 GeV. 

\begin{table} 
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\noalign{\vspace{-8pt}} \hline
            & Their                & Our Fit         & Our  & Our - Their   & Our - Dipole   \\
            &  Fit                 & Their Assum.    & Assum. & $\delta M_A  $& $\delta M_A$  \\ \hline
 Barish 77  & 1.01 $\pm$ 0.09      &1.087 $\pm$ 0.10 & 1.058  & -0.029        & -0.048  \\  
 Miller 82  & 1.05 $\pm$ 0.05      &1.118 $\pm$ 0.055& 1.091 & -0.027        & -0.046  \\
 Kitagaki 83& 1.05$_{-0.16}^{+0.12}$& 1.139$\pm$ 0.10& 1.118 & -0.021        & -0.052   \\ 
 Baker 81   & 1.07 $\pm$ 0.06      & 1.075           & 1.050 &  -0.025       & -0.049   \\ \hline
\end{tabular}
\end{center}
\caption{ Fit values and shifted values of $M_A$ (GeV) from deuterium experiments. 
Column 2 gives the fit values of $M_A$ from
their papers. For Barish and Miller, we give their "shape fit" value,
since this value most closely reflects how we can calculate their $M_A$.
Column 3 gives our fit
value of $M_A$ using their assumptions. Column 4 gives our fit value of 
$M_A$ with our assumptions.
Column 5 gives $\delta M_A$    
between our assumptions minus the experiments assumptions. Column 6 gives $\delta M_A$
between using our form factors and a dipole form factors. For this difference
the value of $g_A$ is kept the constant.}
\label{MA_values}
\end{table}

\begin{figure}
\begin{center}
\epsfxsize=5.01in
\mbox{\epsffile[0 0 567 405]{elas_JhaKJhaJ.eps}}
\end{center}
\caption{ The QE cross section and $\nu$ and $\overline{\nu}$ along 
with data from various experiment. The calculation uses the latest form 
factors and $M_A$ = 1.00 and $G_A$ = -1.267}
\label{elas_JhaKJhaJ}
\end{figure}

Figure~\ref{elas_JhaKJhaJ} shows the QE cross section for 
$\nu$ and $\overline{\nu}$ using our most up to date assumptions.
We have used form factors from cross 
section and  polarization measurements. 
We used $G_A$ = -1.267. We have scaled 
down $M_A$ from the old  best fit of $M_A$=1.026 $\pm$ 0.021 to $M_A$=1.00
(which would have been obtained with the best vector form factors 
known today). 
Even with the most up to date
assumptions on form factors
the agreement between data and
predicction is not spectacular. The data - flux errors are 10\%.
The anti-neutrino data, which is on nuclear targets, is below the 
curves. This is most likely due to nuclear physics effects. Over
the next year, we plan to
 study the nuclear corrections (using models which 
 work in electron scattering~\cite{ejlab}). In addition, we wil 
 be calculating the corrections 
to the $M_A$ values for other experiments
(to update the results for the latest vector form factors). 

\begin{thebibliography}{22}

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\verb+nucl-ex/0204009+.

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\bibitem{LSND}
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\bibitem{numi}\begin{verbatim}
http://www-numi.fnal.gov/fnal_minos/new_initiatives/new_initiatives.html
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\bibitem{eois} 
\begin{verbatim}http://home.fnal.gov/~dharris/xsec_agenda.html\end{verbatim}
%
    
\bibitem{ejlab} JLab experiment 99-118 on the nuclear dependence
of R at low $Q^2$,  A. Brull,
C.E. Keppel spokespersons
\begin{verbatim}(http://www.jlab.org/exp_prog/proposals/99/PR99-118.pdf)\end{verbatim}

 JLab hydrogen experiment 94-110, C.E. Keppel spokesperson
\begin{verbatim}(http://www.jlab.org/exp_prog/proposals/94/PR94-110.pdf)\end{verbatim} JLab deuterium  experiment 02-109, C.E. Keppel, M. E. Christy,
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from 
\begin{verbatim}(http://www.jlab.org/exp_prog/generated/nucleon.html)\end{verbatim}



\end{thebibliography}
\end{document}
